K-Theoretic Equivariant 3-Vertex

• K-theoretic equivariant 3-vertex is a local enumerative building block that defines a canonical trivalent tensor in curve-counting theories on toric threefolds.
• It employs techniques such as equivariant pushforwards, capping operators, and quantum difference equations to transition between bare and capped vertex functions.
• The framework enables effective gluing of local invariants into global partition functions, with significant ties to DT/PT correspondence, mirror symmetry, and quantum toroidal algebras.

Searching arXiv for the core papers and closely related recent work on the K-theoretic equivariant 3-vertex. The K-theoretic equivariant 3-vertex is a local enumerative building block in equivariant $K$-theory for curve-counting theories on toric and related threefolds. In the Donaldson–Thomas/Pandharipande–Thomas setting, it is a canonical 3-valent tensor attached to $X=\mathbb C^3$ or to a corresponding relative geometry, with three legs indexed by partitions or, more canonically, by classes in the equivariant $K$-theory of $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$ (Kononov et al., 2019). In the quasimap and quantum $K$-theoretic literature, closely related “vertex functions” appear as bare and capped generating series, linked by a capping operator that solves a quantum difference equation; this formulation is central for instanton moduli spaces and hypertoric varieties (Smirnov, 2016). Across these settings, the 3-vertex formalism organizes localization, degeneration, gluing, and wall-crossing, and it connects enumerative geometry to quantum toroidal algebras, stable envelopes, and $q$-difference systems.

1. Geometric frameworks and basic objects

In toric $3$-fold Donaldson–Thomas theory, the local model is $X=\mathbb C^3$, and the vertex is the local contribution assigned to the three coordinate directions. Its three legs encode asymptotic partitions of torus-fixed subschemes, and the resulting tensor serves as the local input for global toric partition functions under degeneration and gluing (Kononov et al., 2019). The same trivalent structure persists in the PT formulation, where the moduli spaces parameterize stable pairs

$\mathcal O_X \xrightarrow{s} F$

with $F$ pure $X=\mathbb C^3$0-dimensional and $X=\mathbb C^3$1 $X=\mathbb C^3$2-dimensional (Kononov et al., 2019).

A different but closely connected formulation arises in equivariant quantum $X=\mathbb C^3$3-theory via quasimaps. For the instanton moduli space

$X=\mathbb C^3$4

the basic geometric object is the moduli of framed rank $X=\mathbb C^3$5 torsion-free sheaves on $X=\mathbb C^3$6 with second Chern class $X=\mathbb C^3$7, with framing along the line at infinity. The special case $X=\mathbb C^3$8 places the Hilbert scheme directly inside the vertex formalism (Smirnov, 2016). Here the torus is

$X=\mathbb C^3$9

with framing characters $K$0, tangent weights $K$1, and symplectic scaling $K$2 (Smirnov, 2016).

Hypertoric varieties provide another explicit model of the same general phenomenon. Their vertex functions are defined by $K$3-theoretic counting of quasimaps from $K$4, and the resulting functions satisfy $K$5-difference equations in both Kähler and equivariant variables. In this setting the vertex formalism is not a trivalent tensor in the toric DT sense, but a system of quasimap-counting generating series with the same structural roles of localization, capping, and parameter exchange under mirror symmetry (Smirnov et al., 2020).

2. Bare vertices, capped vertices, and trivalent tensors

For quasimaps with a descendent insertion $K$6, the bare vertex and capped vertex are generating series of equivariant pushforwards from quasimap moduli spaces. Their defining relation is

$K$7

where $K$8 is the capping operator (Smirnov, 2016). Geometrically, $K$9 records the contribution of the open quasimap moduli, $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$0 is the rubber or relative contribution, and $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$1 is the fully capped invariant (Smirnov, 2016).

The capping operator is the fundamental solution of the $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$2-theoretic quantum difference equation

$\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$3

with $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$4 acting by multiplication by the tautological line bundle and $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$5 a difference operator with rational matrix coefficients (Smirnov, 2016). This is the $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$6-theoretic analogue of the quantum differential equation in quantum cohomology.

In toric DT/PT theory, the local vertex is defined directly as a trivalent generating function. For prescribed boundary conditions $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$7 and $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$8, the $\operatorname{Hilb}(\mathbb C^2,\mathrm{points})$9-theoretic vertex is

$K$0

where the moduli spaces carry symmetric perfect obstruction theories $K$1 (Kuhn et al., 2023). The theory uses the Nekrasov–Okounkov twist

$K$2

and the twist is emphasized as essential for rigidity and localization in equivariant $K$3-theory (Kuhn et al., 2023).

These two viewpoints are structurally parallel. The quasimap formalism packages relative geometry into a capping operator, while the toric DT/PT formalism packages local relative conditions into a trivalent tensor. In both cases, the vertex is the local object from which global invariants are assembled.

3. Leg structure and explicit formulas

The leg decomposition is one of the most concrete aspects of the theory. A 2-leg vertex means that exactly two of the three asymptotic partitions are nonempty and one is empty. In the PT-based $K$4-theoretic DT theory of toric $K$5-folds, this case admits an explicit formula in a gauge adapted to

$K$6

with relative divisors $K$7 and torus

$K$8

(Kononov et al., 2019). The result is expressed plethystically as

$K$9

together with an equivalent exponential expansion over all $q$0 (Kononov et al., 2019). The $q$1-leg specialization is obtained by taking the constant term in the $q$2-variables, and its proof uses stable envelopes (Kononov et al., 2019).

A more recent contour-integral formulation computes both the equivariant DT and PT $q$3-vertices from the same JK-residue integrand. The Witten index of a $q$4d $q$5 SQM is written as

$q$6

and the distinction between DT and PT is made solely by the reference vector of the JK prescription: $q$7 extracts the DT poles, while $q$8 extracts the PT poles (Kimura et al., 16 Aug 2025). In this framework, one-leg and two-leg PT cases involve only simple poles, whereas the genuine three-leg PT vertex exhibits second-order poles (Kimura et al., 16 Aug 2025).

The same contour formalism analyzes three asymptotic limits of the PT vertex. The limit $q$9 recovers the normalized unrefined topological vertex; an anisotropic scaling limit yields the refined topological vertex; and a distinct scaling of $3$0 produces the Macdonald refined topological vertex (Kimura et al., 16 Aug 2025). This places the $3$1-theoretic equivariant PT vertex as a common refinement of several established vertex formalisms.

4. Factorization, $3$2-difference equations, and rationality

A central structural result in the capped-vertex literature is factorization under degeneration of framing weights. If the framing torus splits $3$3 and the second block of weights is scaled by a parameter $3$4, then the capping operator satisfies

$3$5

where $3$6 is a universal lower-triangular operator coming from the quantum toroidal algebra representation (Smirnov, 2016). The bare vertex factorizes compatibly in the same limit. Combined with a large-rank classicality statement, this yields the rationality theorem that the $3$7-theoretic $3$8-leg capped descendent vertex is the Taylor expansion of a rational function in

$3$9

(Smirnov, 2016).

For hypertoric varieties, the modified bare vertex satisfies two commuting families of $X=\mathbb C^3$0-difference equations. Circuits of the hyperplane arrangement control the Kähler-shift equations, and cocircuits control the equivariant-shift equations (Smirnov et al., 2020). Under $X=\mathbb C^3$1d mirror symmetry, these two systems are exchanged after swapping Kähler and equivariant variables and reversing the polarization; the mirror transformation matrix is built from elliptic stable envelopes (Smirnov et al., 2020). This makes the vertex function simultaneously a solution to dual $X=\mathbb C^3$2-difference systems and a bridge between mirror pairs.

The same rationality pattern reappears in the study of the index vertex for $X=\mathbb C^3$3. There the index vertex is identified, for big enough generic slopes, with the equivariant Euler characteristic of a twisted $X=\mathbb C^3$4-theoretic stable envelope after applying the $X=\mathbb C^3$5d mirror map, and it follows that the index vertex is the power series expansion of a rational function (Dinkins et al., 2021). A plausible implication is that rationality is not an isolated feature of special toric calculations, but part of a wider stable-envelope mechanism in equivariant $X=\mathbb C^3$6-theoretic vertex theory.

5. Gluing, DT/PT correspondence, and dimensional reduction

In toric $X=\mathbb C^3$7-fold geometry, global partition functions factorize into edge and vertex contributions. For a toric $X=\mathbb C^3$8-fold $X=\mathbb C^3$9,

$\mathcal O_X \xrightarrow{s} F$0

and likewise for PT, with the same edge terms $\mathcal O_X \xrightarrow{s} F$1 in both theories (Kuhn et al., 2023). This concentrates the distinction between DT and PT entirely in the vertices.

The $\mathcal O_X \xrightarrow{s} F$2-fold $\mathcal O_X \xrightarrow{s} F$3-theoretic DT/PT vertex correspondence states

$\mathcal O_X \xrightarrow{s} F$4

and it implies the global toric identity

$\mathcal O_X \xrightarrow{s} F$5

for smooth toric $\mathcal O_X \xrightarrow{s} F$6-folds (Kuhn et al., 2023). The proof uses wall-crossing in a family of weak stability conditions and is given in two versions, one following Mochizuki and one following Joyce. A major technical ingredient is the construction of symmetric almost-perfect obstruction theories on auxiliary master spaces via symmetrized pullback (Kuhn et al., 2023).

The $\mathcal O_X \xrightarrow{s} F$7-leg vertex admits a concrete gluing application to the resolved conifold: gluing two $\mathcal O_X \xrightarrow{s} F$8-leg vertices produces a reduced operator whose vacuum factor is

$\mathcal O_X \xrightarrow{s} F$9

and any matrix element divided by the vacuum matrix element is polynomial in the Kähler parameter $F$0 (Kononov et al., 2019).

A $F$1-fold extension clarifies how the familiar $F$2-vertex sits inside a larger formalism. For toric Calabi–Yau $F$3-folds, the local factors $F$4 and $F$5 specialize to the standard toric $F$6-fold vertex and edge after setting

$F$7

and restricting to fixed loci scheme-theoretically supported on the hyperplane $F$8 (Cao et al., 2019). More precisely,

$F$9

with the analogous PT statement (Cao et al., 2019). This gives an explicit dimensional reduction from the $X=\mathbb C^3$00-fold vertex formalism to the $X=\mathbb C^3$01-theoretic equivariant $X=\mathbb C^3$02-vertex.

6. Algebraic structures underlying the vertex

The $X=\mathbb C^3$03-theoretic equivariant 3-vertex is closely controlled by representation theory. For instanton moduli,

$X=\mathbb C^3$04

where each factor is a Fock space and the framing parameters $X=\mathbb C^3$05 are the evaluation parameters of the corresponding Fock representations (Smirnov, 2016). In this formulation, the capping operator is tied to the fundamental solution of the quantum difference equation, and its factorization is governed by the universal $X=\mathbb C^3$06-matrix, Heisenberg subalgebras, stable envelopes, and the $X=\mathbb C^3$07 equation (Smirnov, 2016).

In the $X=\mathbb C^3$08-leg PT theory, the Hilbert-scheme/symmetric-function correspondence and stable envelopes provide the computational mechanism. The paper uses

$X=\mathbb C^3$09

identifies irreducibles with Schur functions, and relates stable envelopes to symmetric functions through the BKR/Haiman correspondence and Cherednik-algebra/Verma-module technology (Kononov et al., 2019). This is what makes the explicit plethystic formula and the skew-Schur-function description of the $X=\mathbb C^3$10-leg vertex possible.

The contour-integral approach extends this algebraic picture by constructing an operator version of the PT vertex, termed the Pandharipande–Thomas $X=\mathbb C^3$11-character. In this language, the residues are rewritten as free-field correlators involving bosonic operators $X=\mathbb C^3$12, $X=\mathbb C^3$13, and $X=\mathbb C^3$14, and the resulting operator series is linked to the shifted quantum toroidal $X=\mathbb C^3$15 / quiver $X=\mathbb C^3$16-algebra framework (Kimura et al., 16 Aug 2025). In one- and two-leg cases the PT $X=\mathbb C^3$17-character commutes with modified screening charges, while in the three-leg PT case second-order poles force derivative terms such as

$X=\mathbb C^3$18

to appear (Kimura et al., 16 Aug 2025). This clarifies that the three-leg vertex is not merely a combinatorial generating function but also an operator-valued object with nontrivial algebraic singularity structure.

7. Terminological boundaries and related but distinct usages

The phrase “3-vertex” can be misleading because several neighboring literatures use superficially similar language for different objects. In toric DT/PT theory it refers to a trivalent local tensor with three asymptotic legs (Kononov et al., 2019). In quasimap theory it refers more loosely to bare and capped vertex functions, often with one-leg emphasis, whose structure is controlled by a capping operator and $X=\mathbb C^3$19-difference equations (Smirnov, 2016). These are part of the same broad enumerative framework, but they are not identical constructions.

By contrast, torus-equivariant $X=\mathbb C^3$20-point genus-$X=\mathbb C^3$21 $X=\mathbb C^3$22-theoretic Gromov–Witten invariants of flag manifolds are not the $X=\mathbb C^3$23-theoretic equivariant 3-vertex. They concern invariants

$X=\mathbb C^3$24

in quantum $X=\mathbb C^3$25-theory of $X=\mathbb C^3$26 or $X=\mathbb C^3$27, together with a divisor-axiom replacement based on the equivariant quantum Chevalley formula and the quantum Bruhat graph (Lenart et al., 22 May 2025). The common numeral “3” here refers to the number of marked points, not to a trivalent vertex tensor.

Likewise, three-parameter equivariant $X=\mathbb C^3$28-theoretic constructions in exotic geometry should not be confused with the 3-vertex formalism. In the construction of affine quantum Schur algebras of type $X=\mathbb C^3$29 and affine iquantum groups of type AIII, the three parameters arise from a $X=\mathbb C^3$30-equivariant action on Kato’s exotic representation

$X=\mathbb C^3$31

and the paper explicitly notes that this is not a standard “three-vertex” in topological vertex theory (Luo et al., 14 Jun 2026). The shared prefix “three-” there refers to torus factors and parameters, not to a trivalent enumerative tensor.

Taken together, these distinctions show that the modern literature uses “vertex” in a family of related but non-identical senses. The most stable core meaning of the K-theoretic equivariant 3-vertex remains the local equivariant $X=\mathbb C^3$32-theoretic building block for toric $X=\mathbb C^3$33-fold curve counting, together with its capped, $X=\mathbb C^3$34-difference, mirror-symmetric, and operator-theoretic avatars.